PlanetMath: hyperbolic sine integral

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hyperbolic sine integral

(Definition)

The function hyperbolic sine integral (in Latin sinus hyperbolicus integralis) from $\mathbb{R}$ to $\mathbb{R}$ is defined as

$\displaystyle {\mathrm{Shi}}{x} \,:=\, \int_0^x\frac{\sinh t}{t}\,dt,$

(1)

or alternatively as

$\displaystyle {\mathrm{Shi}}{x} \,:=\, \int_0^1\frac{\sinh{tx}}{t}\,dt.$

$\displaystyle {\mathrm{Shi}}{z} = z\!+\!\frac{z^3}{3\!\cdot\!3!}\!+\!\frac{z^5}{5\!\cdot\!5!} +\!\frac{z^7}{7\!\cdot\!7!}\!+\cdots,$

which converges for all complex values

and thus defines an entire transcendental function. Using the Taylor expansions, it is easily seen that

$\displaystyle {\mathrm{Shi}}x \;=\; i\,{\mathrm{Si}}{ix}$

connects Shi to the sine integral function.

${\mathrm{Shi}}{x}$ satisfies the linear third order differential equation

$\displaystyle xf'''(x)\!+\!2f''(x)\!-\!xf'(x) = 0.$

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30A99 (Functions of a complex variable :: General properties :: Miscellaneous)

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